A robust methodology for kinetic model parameter estimation for biocatalytic reactions

Effective estimation of parameters in biocatalytic reaction kinetic expressions are very important when building process models to enable evaluation of process technology options and alternative biocatalysts. The kinetic models used to describe enzyme‐catalyzed reactions generally include several parameters, which are strongly correlated with each other. State‐of‐the‐art methodologies such as nonlinear regression (using progress curves) or graphical analysis (using initial rate data, for example, the Lineweaver‐Burke plot, Hanes plot or Dixon plot) often incorporate errors in the estimates and rarely lead to globally optimized parameter values. In this article, a robust methodology to estimate parameters for biocatalytic reaction kinetic expressions is proposed. The methodology determines the parameters in a systematic manner by exploiting the best features of several of the current approaches. The parameter estimation problem is decomposed into five hierarchical steps, where the solution of each of the steps becomes the input for the subsequent step to achieve the final model with the corresponding regressed parameters. The model is further used for validating its performance and determining the correlation of the parameters. The final model with the fitted parameters is able to describe both initial rate and dynamic experiments. Application of the methodology is illustrated with a case study using the ω‐transaminase catalyzed synthesis of 1‐phenylethylamine from acetophenone and 2‐propylamine. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Parameter determination and validation for a mechanistic model of the enzymatic saccharification of cellulose‐Iβ

Cost‐effective production of fuels and chemicals from lignocellulosic biomass often involves enzymatic saccharification, which has been the subject of intense research and development. Recently, a mechanistic model for the enzymatic saccharification of cellulose has been developed that accounts for distribution of cellulose chain lengths, the accessibility of insoluble cellulose to enzymes, and the distinct modes of action of the component cellulases [Griggs et al. (2012) Biotechnol. Bioeng., 109(3):665–675; Griggs et al. (2012) Biotechnol. Bioeng., 109(3):676–685]. However, determining appropriate values for the adsorption, inhibition, and rate parameters required further experimental investigation. In this work, we performed several sets of experiments to aid in parameter estimation and to quantitatively validate the model. Cellulosic materials differing in degrees of polymerization and crystallinity (α‐cellulose‐I_β and highly crystalline cellulose‐I_β) were digested by component enzymes (EG_I/CBH_I/ ) and by mixtures of these enzymes. Based on information from the literature and the results from these experiments, a single set of model parameters was determined, and the model simulation results using this set of parameters were compared with the experimental data of total glucan conversion, chain‐length distribution, and crystallinity. Model simulations show significant agreement with the experimentally derived glucan conversion and chain‐length distribution curves and provide interesting insights into multiple complex and interacting physico‐chemical phenomena involved in enzymatic hydrolysis, including enzyme synergism, substrate accessibility, cellulose chain length distribution and crystallinity, and inhibition of cellulases by soluble sugars. © 2015 American Institute of Chemical Engineers Biotechnol. Prog., 31:1237–1248, 2015     

Mechanistic kinetic model for symmetric carboligations using benzaldehyde lyase

For reactions using thiamine diphosphate (ThDP)-dependent enzymes many empirically-derived kinetic models exist. However, there is a lack of mechanistic kinetic models. This is especially true for the synthesis of symmetric 2-hydroxy ketones from two identical aldehydes, with one substrate acting as the donor and the other as the acceptor. In this contribution, a systematic approach for deriving such a kinetic model for thiamine diphosphate (ThDP)-dependent enzymes is presented. The derived mechanistic kinetic model takes this donor-acceptor principle into account by containing two K(m)-values even for identical substrate molecules. As example the stereoselective carbon-carbon coupling of two 3,5-dimethoxy-benzaldehyde molecules to (R)-3,3',5,5'-tetramethoxy-benzoin using benzaldehyde lyase (EC 4.1.2.38) from Pseudomonas fluorescens is studied. The model is derived using a model-based experimental analysis method which includes parameter estimation, model analysis, optimal experimental design, in silico experiments, sensitivity analysis and model revision. It is shown that this approach leads to a robust kinetic model with accurate parameter estimates and an excellent prediction capability.     

Straightforward method for calibration of mechanistic cation exchange chromatography models for industrial applications

Mechanistic modeling of chromatography processes is one of the most promising techniques for the digitalization of biopharmaceutical process development. Possible applications of chromatography models range from in silico process optimization in early phase development to in silico root cause investigation during manufacturing. Nonetheless, the cumbersome and complex model calibration still decelerates the implementation of mechanistic modeling in industry. Therefore, the industry demands model calibration strategies that ensure adequate model certainty in a limited amount of time. This study introduces a directed and straightforward approach for the calibration of pH-dependent, multicomponent steric mass action (SMA) isotherm models for industrial applications. In the case investigated, the method was applied to a monoclonal antibody (mAb) polishing step including four protein species. The developed strategy combined well-established theories of preparative chromatography (e.g. Yamamoto method) and allowed a systematic reduction of unknown model parameters to 7 from initially 32. Model uncertainty was reduced by designing two representative calibration experiments for the inverse estimation of remaining model parameters. Dedicated experiments with aggregate-enriched load material led to a significant reduction of model uncertainty for the estimates of this low-concentrated product-related impurity. The model was validated beyond the operating ranges of the final unit operation, enabling its application to late-stage downstream process development. With the proposed model calibration strategy, a systematic experimental design is provided, calibration effort is strongly reduced, and local minima are avoided.     

Parametric sensitivity analysis and reduction of a detailed nutritional model of plant cell cultures

A kinetic model of plant nutrition described by Cloutier et al. (Cloutier et al., 2008. Biotechnol Bioeng 99:189-200) is progressively simplified so as to obtain a predictive model that describes the evolution of the biomass and the extracellular and intracellular concentrations of three determining nutrients, that is, free intracellular nitrogen, phosphate, and carbohydrate compounds. Three techniques of global sensitivity analysis are successively applied to assess the model parameter influence and potential correlation. The resulting dynamic model is able to predict plant growth for the two most encountered plant bioprocesses, namely suspension cells and hairy roots.     

Making sense of parameter estimation and model simulation in bioprocesses

Most articles that report fitted parameters for kinetic models do not include meaningful statistical information. This study demonstrates the importance of reporting a complete statistical analysis and shows a methodology to perform it, using functionalities implemented in computational tools. As an example, alginate production is studied in a batch stirred-tank fermenter and modeled using the kinetic model proposed by Klimek and Ollis (1980). The model parameters and their 95% confidence intervals are estimated by nonlinear regression. The significance of the parameters value is checked using a hypothesis test. The uncertainty of the parameters is propagated to the output model variables through prediction intervals, showing that the kinetic model of Klimek and Ollis (1980) can simulate with high certainty the dynamic of the alginate production process. Finally, the results obtained in other studies are compared to show how the lack of statistical analysis can hold back a deeper understanding about bioprocesses.     

An evolutionary computing approach for parameter estimation investigation of a model for cholera

We consider the problem of using time-series data to inform a corresponding deterministic model and introduce the concept of genetic algorithms (GA) as a tool for parameter estimation, providing instructions for an implementation of the method that does not require access to special toolboxes or software. We give as an example a model for cholera, a disease for which there is much mechanistic uncertainty in the literature. We use GA to find parameter sets using available time-series data from the introduction of cholera in Haiti and we discuss the value of comparing multiple parameter sets with similar performances in describing the data.     

Steady-state kinetics of bifunctional enzymes. Taking into account kinetic hierarchy of fast and slow catalytic cycles in a generalized model

A steady-state approximation of the generalized two-dimensional model of a bifunctional enzyme catalyzing independent proceeding of two one-pathway reactions is considered in a case of mutual influence of the active sites. Coexistence of fast and slow catalytic cycles in the reaction mechanism is analyzed. Conditions when the hierarchy of fast and slow catalytic cycles allows simplification of a two-dimensional model and its reduction to the one-dimensional cyclic schemes were determined. Kinetic equations describing these simplified schemes are presented.     

A reinforcement learning-based hybrid modeling framework for bioprocess kinetics identification

Constructing predictive models to simulate complex bioprocess dynamics, particularly time-varying (i.e., parameters varying over time) and history-dependent (i.e., current kinetics dependent on historical culture conditions) behavior, has been a longstanding research challenge. Current advances in hybrid modeling offer a solution to this by integrating kinetic models with data-driven techniques. This article proposes a novel two-step framework: first (i) speculate and combine several possible kinetic model structures sourced from process and phenomenological knowledge, then (ii) identify the most likely kinetic model structure and its parameter values using model-free Reinforcement Learning (RL). Specifically, Step 1 collates feasible history-dependent model structures, then Step 2 uses RL to simultaneously identify the correct model structure and the time-varying parameter trajectories. To demonstrate the performance of this framework, a range of in-silico case studies were carried out. The results show that the proposed framework can efficiently construct high-fidelity models to quantify both time-varying and history-dependent kinetic behaviors while minimizing the risks of over-parametrization and over-fitting. Finally, the primary advantages of the proposed framework and its limitation were thoroughly discussed in comparison to other existing hybrid modeling and model structure identification techniques, highlighting the potential of this framework for general bioprocess modeling.     

Substrate uptake,phosphorus repression,and effect of seed culture on glycopeptide antibiotic production: Process model development and experimental validation

Actinomycetes, the soil borne bacteria which exhibit filamentous growth, are known for their ability to produce a variety of secondary metabolites including antibiotics. Industrial scale production of such antibiotics is typically carried out in a multi‐substrate medium where the product formation may experience catabolite repression by one or more of the substrates. Availability of reliable process models is a key bottleneck in optimization of such processes. Here we present a structured kinetic model to describe the growth, substrate uptake and product formation for the glycopeptide antibiotic producer strain Amycolatopsis balhimycina DSM5908. The model is based on the premise that the organism is an optimal strategist and that the various metabolic pathways are regulated via key rate limiting enzymes. Further, the model accounts for substrate inhibition and catabolite repression. The model is also able to predict key phenomena such as simultaneous uptake of glucose and glycerol but with different specific uptake rates, and inhibition of glycopeptide production by high intracellular phosphate levels. The model is successfully applied to both production and seed medium with varying compositions and hence has good predictive ability over a variety of operating conditions. The model parameters are estimated via a well‐designed experimental plan. Adequacy of the proposed model was established via checking the model sensitivity to its parameters and confidence interval calculations. The model may have applications in optimizing seed transfer, medium composition, and feeding strategy for maximizing production. Biotechnol. Bioeng. 2010;105: 109–120. © 2009 Wiley Periodicals, Inc.     

Optimization-driven identification of genetic perturbations accelerates the convergence of model parameters in ensemble modeling of metabolic networks

The ensemble modeling (EM) approach has shown promise in capturing kinetic and regulatory effects in the modeling of metabolic networks. Efficacy of the EM procedure relies on the identification of model parameterizations that adequately describe all observed metabolic phenotypes upon perturbation. In this study, we propose an optimization-based algorithm for the systematic identification of genetic/enzyme perturbations to maximally reduce the number of models retained in the ensemble after each round of model screening. The key premise here is to design perturbations that will maximally scatter the predicted steady-state fluxes over the ensemble parameterizations. We demonstrate the applicability of this procedure for an Escherichia coli metabolic model of central metabolism by successively identifying single, double, and triple enzyme perturbations that cause the maximum degree of flux separation between models in the ensemble. Results revealed that optimal perturbations are not always located close to reaction(s) whose fluxes are measured, especially when multiple perturbations are considered. In addition, there appears to be a maximum number of simultaneous perturbations beyond which no appreciable increase in the divergence of flux predictions is achieved. Overall, this study provides a systematic way of optimally designing genetic perturbations for populating the ensemble of models with relevant model parameterizations.     

A mechanistic mathematical model for the catalytic action of glutathione peroxidase

Abstract

Glutathione peroxidase (GPx) is a well-known seleno-enzyme that protects cells from oxidative stress (e.g., lipid peroxidation and oxidation of other cellular proteins and macromolecules), by catalyzing the reduction of harmful peroxides (e.g., hydrogen peroxide: H₂O₂) with reduced glutathione (GSH). However, the catalytic mechanism of GPx kinetics is not well characterized in terms of a mathematical model. We developed here a mechanistic mathematical model of GPx kinetics by considering a unified catalytic scheme and estimated the unknown model parameters based on different experimental data from the literature on the kinetics of the enzyme. The model predictions are consistent with the consensus that GPx operates via a ping-pong mechanism. The unified catalytic scheme proposed here for GPx kinetics clarifies various anomalies, such as what are the individual steps in the catalytic scheme by estimating their associated rate constant values and a plausible rationale for the contradicting experimental results. The developed model presents a unique opportunity to understand the effects of pH and product GSSG on the GPx activity under both physiological and pathophysiological conditions. Although model parameters related to the product GSSG were not identifiable due to lack of product-inhibition data, the preliminary model simulations with the assumed range of parameters show that the inhibition by the product GSSG is negligible, consistent with what is known in the literature. In addition, the model is able to simulate the bi-modal behavior of the GPx activity with respect to pH with the pH-range for maximal GPx activity decreasing significantly as the GSH levels decrease and H₂O₂ levels increase (characteristics of oxidative stress). The model provides a key component for an integrated model of H₂O₂ balance under normal and oxidative stress conditions.     

Mass-transfer limitations for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor

A mathematical model has been developed for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor in which the enzyme-catalyzed reaction (the irreversible uni-uni competitive Michaelis-Menten kinetics is chosen as an example) was coupled with intraparticle diffusion, external mass transfer, and axial dispersion. The effects of mass-transfer limitations, competitive inhibition of substrates, deactivation on the enzyme effective enantioselectivity, and the optical purity and yield of the desired product are examined quantitatively over a wide range of parameters using the orthogonal collocation method. For a first-order reaction, an analytical solution is derived from the mathematical model for slab-, cylindrical-, and spherical-enzyme supports. Based on the analytical solution for the steady-state resolution process, a new concise formulation is presented to predict quantitatively the mass-transfer limitations on enzyme effective enantioselectivity and optical purity and yield of the desired product for a continuous steady-state kinetic resolution process in a fixed-bed reactor.     

Scalar modeling and analysis of a 3D biochemical reaction model

For many systems it is advantageous if analysis and modeling can be accomplished from a scalar time series because this greatly facilitates the experimental setup. Moreover, in real-life systems it is hardly true that all the state variables are available for analysis and modeling. Since the late 1980s, techniques have been put forward for building mathematical models from a scalar time series. One of the objectives of this paper is to verify if it is possible to obtain global non-linear models (non-linear differential equations) from scalar time series. Such data are obtained using a model of biochemical reaction with aperiodic (chaotic) oscillations as recently observed in the case of a glycolytic reaction (Nielsen, K., Sorensen, P.G., Hynne, F., 1997. Chaos in glycolysis. J Theor. Biol. 186, 303-306.). The main objective, however, is to investigate which state variable is more convenient for the task in practice. It is shown that observability indices seem to quantify quite well which variable should be preferred as the observable. The validity of the results are established performing rigorous topological analysis on the original system and the obtained models. The influence of noise, always present in experimental time series, on the dynamics underlying such a system is also investigated.     

System identification and parameter estimation in mathematical medicine: examples demonstrated for prostate cancer

We review our studies on how to identify the most appropriate models of diseases, and how to determine their parameters in a quantitative manner given a short time series of biomarkers, using intermittent androgen deprivation therapy of prostate cancer as an example. Recently, it has become possible to estimate the specific parameters of individual patients within a reasonable time by employing the information concerning other previous patients as a prior. We discuss the importance of using multiple mathematical methods simultaneously to achieve a solid diagnosis and prognosis in the future practice of personalized medicine.     

Structure and function of a novel endonuclease acting on branched DNA substrates

We show that Pyrococcus abyssi PAB2263 (dubbed NucS (nuc lease for s s DNA) is a novel archaeal endonuclease that interacts with the replication clamp PCNA. Structural determination of P. abyssi NucS revealed a two‐domain dumbbell‐like structure that in overall does not resemble any known protein structure. Biochemical and structural studies indicate that NucS orthologues use a non‐catalytic ssDNA‐binding domain to regulate the cleavage activity at another site, thus resulting into the specific cleavage at double‐stranded DNA (dsDNA)/ssDNA junctions on branched DNA substrates. Both 3′ and 5′ extremities of the ssDNA can be cleaved at the nuclease channel that is too narrow to accommodate duplex DNA. Altogether, our data suggest that NucS proteins constitute a new family of structure‐specific DNA endonucleases that are widely distributed in archaea and in bacteria, including Mycobacterium tuberculosis.